Decision Feedback Equalization (DFE) is a well-known scheme used to detect signals transmitted across communications and recording channels with intersymbol interference. In DFE, a forward equalizer shapes the readback pulse in a desired way and a feedback equalizer attempts to cancel the intersymbol interference. However, a problem arises with DFE, in that the feedback mechanism has the potential to cause long bursts of errors.
The version of DFE called Multilevel Decision Feedback Equalization (MDFE), described by J. G. Kenney, L. R. Carley and R. Wood in the article "Multi-level Decision Feedback Equalization for Saturation Recording," EEE Transaction on Magnetics, vol. 19, no. 4, pp. 2160-71, July 1993, has several advantages when used in magnetic recording. In MDFE, the channel is equalized into a linear channel on non-return to zero (NRZ) (i.e. .+-.1) sequences defined by dibit response with a main lobe of positive taps a.sub.0, a.sub.1, a.sub.2 and a tail of negative taps -a.sub.3, -a.sub.4, -a.sub.5, . . . , -a.sub.L. The feedback equalizer attempts to cancel the negative taps. According to the formulation by J. L. Kenney and C. M. Melas in the article "Pipelining for Speed Doubling in MDFE," Conference Proceedings of the ICC '96, it is required that a.sub.1 &gt;a.sub.0 =a.sub.2, yielding signal levels .+-.(a.sub.1 +2a.sub.0), .+-.a.sub.1, .+-.(a.sub.1 -2a.sub.0). After cancellation, a threshold detector, with threshold =0, makes the bit decision. This would not work if the lowest level, a.sub.1 -2a.sub.0, were very small or negative, so the data sequences are constrained to eliminate that level. This is accomplished by imposing a d=1 run length limited constraint (which forbids the NRZ patterns 1 -1 1 and -1 1 -1). An upper run length limit is also imposed for purposes of timing control. Typically a standard rate 2:3 (d, k)=(1, 7) run length code is used.
For many sets of taps, a standard (1, 7) code, such as that described in U.S. Pat. No. 4,413,251 to Adler et al., admit infinitely propagating error sequences. These are sequences that, as a result of an initial error, can cause infinitely many errors, even in the absence of further noise. Such sequences are among the primary causes of error rate degradation in the MDFE scheme, especially in the region of high signal-to-noise ratio. A need arises for a scheme that eliminates all infinitely propagating error sequences for many sets of taps.